Q.State whether the following statement is True or False: In a LPP, the maximum value of the objective function $Z = ax + by$ is always finite.

The key idea is that a Linear Programming Problem may have an unbounded feasible region, which can lead to an unbounded objective function if the region extends infinitely in the direction of optimisation.

Reasoning:

1. For the maximum value of $Z$ to be finite, the feasible region must be bounded in the direction of the objective function's gradient.
2. If the feasible region is unbounded and the objective function can increase indefinitely along a feasible direction, the maximum is not finite. …

Why This Statement is Tricky

Many students assume that every LPP must have a finite optimal value because they only work with bounded feasible regions in textbook problems. But the key lies in the nature of the feasible region itself.

The objective function $Z = ax + by$ is a linear function. Its maximum over a region depends entirely on whether the region "closes in" on all sides or stretches out to infinity.

Step-by-Step Reasoning

1. The feasible region can be bounded or unbounded.

A bounded region is enclosed within a finite area (like a polygon). An unbounded region extends infinitely in at least one direction — think of a half-plane or a wedge that never closes.

2. When the region is bounded, the maximum is always finite.

This is a direct consequence of the Extreme Value Theorem for continuous functions on a closed, bounded set. Since $Z$ is linear (and therefore continuous), and the feasible region is a closed, bounded polygon, $Z$ attains both a maximum and a minimum at some corner point. This is the standard result taught in most LPP chapters.

3. When the region is unbounded, the maximum may be infinite.

Consider a simple example:

Maximize $Z = x + y$ subject to $x \geq 0, y \geq 0$.

The feasible region is the entire first quadrant — it goes to infinity along both axes. As $x$ and $y$ increase, $Z$ grows without bound. There is no finite maximum.

4. But an unbounded region does not guarantee an infinite maximum.

Here’s the nuance: even if the region is unbounded, the objective function might still have a finite maximum if the direction of increase is "blocked" by constraints.

Example:

Maximize $Z = x + y$ subject to $x \geq 0, y \geq 0, x + y \leq 10$.

The region is unbounded? Actually no — the third constraint bounds it. But consider:

Maximize $Z = -x - y$ subject to $x \geq 0, y \geq 0$.

Here the region is unbounded, but $Z$ is always negative or zero, and its maximum is $0$ (at $x=0, y=0$), which is finite. So the statement fails in both directions — it claims the maximum is always finite, but we have counterexamples where it is not. …

Method: Deciding whether a finite optimum exists

Whenever a claim says an LPP optimum is "always finite," the technique is to separate two independent facts: is the region bounded? and does the objective grow in the open direction?

Steps

Step 1: Classify the feasible region.

Bounded (a closed polygon) $\Rightarrow$ by the Extreme Value Theorem both max and min are finite and sit at corners. Unbounded $\Rightarrow$ an optimum may fail to exist — but not automatically.

Step 2: Test the open direction against the objective.

For a maximum over an unbounded region, check the half-plane $ax+by>M$ (with $M$ the best corner value). If it meets the region, $Z$ has no finite maximum; if it misses, $M$ is the maximum. …

Common Mistakes

Mistake 1: Assuming every LPP has a finite optimum.

Why it's wrong: over an unbounded region the objective can increase without bound (e.g. maximise $Z=x+y$ on the whole first quadrant). Correct approach: classify the region as bounded or unbounded before claiming a finite maximum.

Mistake 2: Assuming an unbounded region always gives an infinite maximum. …